This group is given on p.215 of Burnside, where with Hudson's group and the alternating group it forms the only three primitive groups of degree 6. It is presented in the form of three generating transpositions: 265341, 234516 and 135246, which are of cycle sets (3,3), (5,1) and (4,1,1) respectively, obscuring the presence of 6-part transpositions in the group. As with any other mixed group, the positive rows form a sub-group of half the order; this is Hudson's group.

Hudson's group is isomorphic with the rotation group of the icosahedron (or its dual, the dodecahedron). An icosahedron has 12 vertices, in 6 mutually opposite pairs, and if these pairs be labelled 1 to 6, the rotations transpose the figures according to Hudson's group. But the group of order 120 is not so easily represented. It has 6-fold cycles, hence its geometrical representation has 20 hexagons, and these may be placed 5 to a vertex giving a multiply-connected surface such as illustrated in Burnside pp.395, 420.

What, however, is the connection with the icosahedron? To discover this, I separated the group into its 60 +ve and 60 -ve rows. The +ve rows, including rounds, formed a sub-group which is Hudson's, and I sketched the corresponding labelling of the icosahedron vertices. The -ve rows (forming a coset) were then transfigured so as to create rounds, thus giving another version of Hudson's rows, and the corresponding icosahedral labelling was plotted. What was the relation between the two plottings?

I contrived the first icosahedron so that, around the 1 vertex, the vertices 2, 3, 4, 5, 6 were in clockwise order. On the other side of the model, they were similarly in anticlockwise order. It was then found that on the second icosahedron the cycles around the 1 vertices were the other two "involute" cycles of 23456, viz 24635 and its reverse 25364, and this was true of every other pair of vertices, 2 to 6, also. The fact that this can be done with the icosahedron is equivalent to the existence of the group of order 120. Thus, the group may be represented by a set of two labelled icosahedra.

It follows that the group of 120 may be formed from Hudson's by involuting the 5-part cycles. Referring to the listing of Hudson's group, each column has a fixed bell leading and there are two mutually reverse cycles of 5; the first column being the dihedral group of order 10, and the other columns its cosets. If the other two involute cycles be added to each column, the group order 120 results. However, this does not reveal that 6-fold cycles are present.

A feature of the group of rows is that any permutation of three of the six figures, such as 623, occurs in one of the 6-part cycles only. There are 120 such permutations, and the 20 6-part cycles each have 6 of them. It follows from this that, if in the 120 rows of the group any fixed trio of figures be permuted (6 arrangements), the extent of 720 will be generated. This suggests 6-bell principles in which the 120 rows of the plain course constitute the group [6.03], and calls permuting any three bells would give the extent. Exhaustive searches showed that there is only one such principle, which was rung as Pitman Minor in 1951. See RW 1970 pp. 474, 514, 614 and the Central Council Collection of rung principles. The pure form of the principle is a 6-part course as follows, by the places:

-36-14.36-36.14.36-36.14.36-36.14-36-14

In this pure form the rows of the plain course constitute the group [6.03]. However, the simplicity of permuting three bells for the extent makes it possible to vary the final 14, and the published version has 16 with 14 as Bob, 1456 as Single.

It is interesting to note that this principle inverts the usual role of a group in composition. The rows of the plain course are a group of transpositions and the permuting of three bells must be regarded as a set of transfigures. The cosets are whole courses.

In order to assist in the attempt to find all possible permutation groups on up to 7 working bells, random search computer programs were written, using a number of existing programs.

Separate investigations were made for 5, 6 and 7 working bells, as a general program would have been much more involved (it was assumed that all groups on 2, 3, 4 working bells had been enumerated). A start was made by enumerating as many groups as possible, and analysing their transposition cycle sets logically. Algorithms were devised for identifying (a) the category of a transposition by its cycles, and (b) the identifying of a group from its signature. Look-up tables were compiled for each algorithm, the second one for identifying a group being updated as hitherto unlisted groups were discovered. These second logical look-up tables are given above.

The technique devised was:

- To choose at random two, or three, rows from the extent, as generators for a putative sub-group. These generators must differ from rounds and from one another, and there must not be a fixed bell among the generators and rounds.
- To construct a group from the chosen generators, by successive application of their transpositions, starting from rounds. Frequently this will give the extent itself, which is not wanted, so an upper limit of rows is set, after which the process is abandoned and fresh generators are chosen. This upper limit is discussed below.
- Having constructed a sub-group of the extent, each row (apart from rounds) is analysed for its transposition cycle sets, the equivalent of the signature is compiled, and the group is identified. If the new group is not listed, the look-up algorithm records the status quo and execution stops. A condensed record is made on disk of all sub-groups identified, giving their generators, order and identification so that they may be investigated subsequently and a count kept of occurrences.
- Each newly-found group is investigated and catalogued, the second look-up table for the program is revised and existing groups in the catalogue, if following the new entry, are re-numbered.

This process can not be regarded as exhaustive; but it assists towards making the catalogue complete. A number of logical assumptions were made:

- It was assumed that if two groups have the same signature, then they are identical, i.e. that one can be transformed into the other by transposing, transfiguring and/or re-ordering, its rows. No case of different groups with the same signature has yet been discovered, but there is a near-case in [6.14] and [7.33], both isomorphic with the rotation group of the tetrahedron, and differing only in the number of working bells - the transposition cycle sets of [7.33] have an extra fixed bell. Other pairs of solutions show a similar relation: [6.09] and [7.28], [6.35] and [4.04]. It is possible that "new" groups might have been constructed but not identified as distinct, because their signatures and degrees were the same as those of listed groups.
- It was assumed that a sub-group of order half an extent can only be the alternating group (i.e. all the +ve rows). Hence apart from this, one-third of the extent (240 in the case of 6 working bells) was taken as the upper limit before abandoning a particular search, and the alternating group was never reached.
- At first it was assumed that two generators were sufficient to generate a sub-group. Starting with the 6-bell groups this soon produced some hitherto-unlisted groups, but zero frequencies made it clear that some of the already-listed groups could not result from two generators (this was checked directly as being so for some groups) and hence there were possibly some unlisted ones which could not be so found. Subsequently three generators were used. The groups on six bells which need three generators are 6.20, 6.21, 6.30, 6.33, 6.34. Clearly, three generators would not suffice for an 8-bell extension of [6.34]. Recently, Pascal search programs using up to 4 random generators have continued the random generation of groups more efficiently.

Under the group listing, for each group will be found its normal subgroups and supergroups (if any). The finding of the normal subgroups was incorporated in a much more rigorous examination of subgroups by random group generation, which discovered a number of errors in the subgroups quoted in the first edition of this paper (January 1994); notably, 6.14 is a subgroup of 6.05; 6.35 is not a subgroup of 7.12. The new analysis very quickly produced one further group, [7.40]. The normal supergroups are quoted with their orders and parity (mixed or positive) in order to assist the practical use of normality in composition, which is explained in the last section of this paper.

This topic "pairs off" certain groups on up to 6 bells, so that if one of a pair had been found and the other not, the correspondence would identify the other one.

Consider the six figures 1-6 subdivided into three pairs:

(12)(34)(56)

These are to be regarded as combinations, not permutations, in that the particular order within a pair, or the order of the pairs, is irrelevant. Compare with the problem of the number of ways of splitting six people into three games of chess.

The number of possible pairs among 6 figures is 6C2 = 15. We can achieve all 15 such pairs by five sets of 3-pairs, for instance:

(12)(34)(56) (13)(25)(46) (14)(26)(35) (15)(24)(36) (16)(23)(45)

Notice that these have been displayed in such a way that the two figures in each pair are in rising order, in each line the first figures of each pair are in rising order, and (down a column) the second figures in the first pairs are in rising order 2 to 6. This "standard form" makes identification of identical 5-rows easier.

But the above 5-row is not unique. It does not contain the row (12)(35)(46). Any 5-row in standard form must have the following figures:

(12)(3.)(..) (13)(2.)(..) (14)(2.)(..) (15)(2.)(..) (16)(2.)(..)

There are three different ways of completing the first row:

(12)(34)(56) (12)(35)(46) (12)(36)(45) (13)(2x)(..) (13)(2.)(..) (13)(2.)(..) etc etc etc

Now, in the first set x cannot be 4, as this leaves 56 for the final pair, one which has already occurred. So x is either 5 or 6. In this way each of the three has two alternatives, and we have six differently-evolving 5-rows:

(12)(34)(56) (12)(35)(46) (12)(36)(45) (13)(25)(46) (13)(24)(56) (13)(24)(56) (14)(2y)(..) (14)(2.)(..) (14)(2.)(..) (15)(2.)(..) (15)(2.)(..) (15)(2.)(..) (16)(2.)(..) (16)(2.)(..) (16)(2.)(..) (12)(34)(56) (12)(35)(46) (12)(36)(45) (13)(26)(45) (13)(26)(45) (13)(25)(46) (14)(2.)(..) (14)(2.)(..) (14)(2.)(..) (15)(2.)(..) (15)(2.)(..) (15)(2.)(..) (16)(2.)(..) (16)(2.)(..) (16)(2.)(..)

Now, y in the top-left 5-row might possibly be 3, 5 or 6. But 3 leaves 56 for the final pair, a combination already appearing above; and 25 also has already been used; so y must be 6. By similar logic, there is no choice left and the six possible 5-rows result:

1. (12)(34)(56) 2. (12)(35)(46) 3. (12)(36)(45) (13)(25)(46) (13)(24)(56) (13)(24)(56) (14)(26)(35) (14)(25)(36) (14)(26)(35) (15)(24)(36) (15)(26)(34) (15)(23)(46) (16)(23)(45) (16)(23)(45) (16)(25)(34) 4. (12)(34)(56) 5. (12)(35)(46) 6. (12)(36)(45) (13)(26)(45) (13)(26)(45) (13)(25)(46) (14)(25)(36) (14)(23)(56) (14)(23)(56) (15)(23)(46) (15)(24)(36) (15)(26)(34) (16)(24)(35) (16)(25)(34) (16)(24)(35)

These six 5-rows have been numbered (arbitrarily) 1. to 6. By good fortune, the number of 5-rows corresponds to the number of numbers involved. We now proceed to transform a 5-row by transfiguring its constituent numbers. Take any permutation on 6 bells, for instance 463215:

1 2 3 4 5 6 4 6 3 2 1 5

We regard this as transfigure X, replacing 1 by 4, 2 by 6 etc., and operate with it on 5-row No.1.:

1. (12)(34)(56) (46)(32)(15) (12)(36)(45) 3. (13)(25)(46) X (43)(61)(25) (13)(24)(56) (14)(26)(35) ---> (42)(65)(31) = (14)(26)(35) (15)(24)(36) (41)(62)(35) (15)(23)(46) (16)(23)(45) (45)(63)(21) (16)(25)(34)

Here, the first transformation is a transfigure by X, but the second is merely a rearrangement of pairs within rows, and reordering of rows, to bring the resultant 5-row into standard form. So X transforms 1. into 3.; it will also transform the others.

Thus a transfigure X permutes the six 5-rows among themselves, defining a permutation Y. In the above example, Y would be found to be 362415 (we have already established the 3 in 1sts place). There is a set of relations between the kinds of transfigures that are involved in such a transformation:

Transfigure kind (6) becomes (3,2,1) (5,1) ... (5,1) (4,2) ... (4,2) (4,1,1) ... (4,1,1) (3,3) ... (3,1,1,1) (3,2,1) ... (6) (3,1,1,1) ... (3,3) (2,2,2) ... (2,1,1,1,1) (2,2,1,1) ... (2,2,1,1) (2,1,1,1,1) ... (2,2,2)

Observe that these relations preserve the parity of transfigures.

The net effect is to map the extent on 6 bells onto itself. Rounds maps onto rounds. If a subgroup of the extent be taken and transformed, the result will be a subgroup. All the permutations of a group, except rounds, are transformed, and the resultant group may be the same group, or a different one. This process is not limited to the 6-bell groups; those on 5 bells or less are also involved, by packing the rows with stationary bells to make up to 6. It is interesting to note that the only extent which has an outer isomorphism is that on 6 figures (Burnside p.209).

Even before an actual checking process, an analysis of the logical tables of transposition types reveals much. In the following two tables, all the known permutation groups on 6 or less bells have been assembled, an asterisk denoting that the transposition type of the column is not possible because the group has not enough bells. In the first table, pairs of groups are brought together which have the dual relationship.

+ve cycle sets -ve cycle sets 5, 4, 3, 3, 2, 6, 4, 3, 2, 2, 1 2, 3, 1, 2, 1, 2, 2, 1, 1, 1, 1 1 2 1, Geometrical 1 1 1, representation 1 |---| |-------| |---| 6.03 24 - 20 - 15 20 30 - 10 - | Two 5.01 24 * * 20 15 * 30 20 * 10 | icosahedra 6.05 24 - 20 - 15 - - - - - | Icosahedron 5.02 24 * * 20 15 * - - * - | 6.06 - 6 8 - 9 8 6 - 7 3 6.17 - 6 - 8 9 - 6 8 3 7 6.09 - 6 8 - 9 - - - - - | Cube 6.19 - 6 - 8 9 - - - - - | 6.11 - - 8 - 3 8 - - 1 3 6.18 - - - 8 3 - - 8 3 1 6.10 - - 8 - 3 - 6 - 6 - | Cube 4.01 * * * 8 3 * 6 * * 6 | 6.14 - - 8 - 3 - - - - - | Tetrahedron 4.02 * * * 8 3 * - * * - | 6.08 - - 4 4 9 12 - - 6 - 6.28 - - 4 4 9 - - 12 - 6 6.12 - - 4 4 - 6 - - 3 - | Tessellation 6.29 - - 4 4 - - - 6 - 3 | of hexagons 6.13 - - 2 - 3 2 - - 4 - | Hexagonal 5.06 - * * 2 3 * - 2 * 4 | prism 6.32 - - 2 - 3 - - - - - | Triangular 5.07 - * * 2 3 * - - * - | prism 6.15 - - 2 - - 2 - - 1 - | Hexagon 5.08 - * * 2 - * - 2 * 1 | 6.16 - - 2 - - - - - 3 - | Triangular 3.01 * * * 2 * * * * * 3 | prism 6.33 - - 2 - - - - - - - | Triangle 3.02 * * * 2 * * * * * - | 6.25 - - - - 3 - 2 - 2 - | Square 4.03 * * * - 3 * 2 * * 2 | prism 6.21 - - - - 3 - - - 3 1 6.34 - - - - 3 - - - 1 3 6.35 - - - - 3 - - - - - | Cuboid 4.04 * * * - 3 * - * * - | 6.27 - - - - 1 - - - 2 - | Cuboid 4.06 * * * - 1 * - * * 2 | 6.37 - - - - - - - - 1 - | Rectangle 2.01 * * * * * * * * * 1 |

In the second table following are groups which transform into themselves. [6.01], the extent, must do so; and so must [6.02], the alternating group, as parity is preserved by the transformations.

+ve cycle sets -ve cycle sets 5, 4, 3, 3, 2, 6, 4, 3, 2, 2, 1 2, 3, 1, 2, 1, 2, 2, 1, 1, 1, 1 1 2 1, 1 1 1, 1 |---| |-------| |---| 6.01 144 90 40 40 45 120 90 120 15 15 6.02 144 90 40 40 45 - - - - - 5.03 4 * * - 5 * 10 - * - 5.04 4 * * - 5 * - - * - 5.05 4 * * - - * - - * - 6.04 - 18 4 4 9 12 - 12 6 6 6.07 - 18 4 4 9 - - - - - 6.20 - 2 - - 5 - 2 - 3 3 6.23 - 2 - - 5 - - - - - 6.22 - 2 - - 1 - 2 - 1 1 6.24 - 2 - - 1 - - - 2 2 6.26 - 2 - - 1 - - - - - 6.30 - - 4 4 9 - - - - - 6.31 - - 4 4 - - - - - - 4.05 * * * - 1 * 2 * * - 6.36 - - - - 1 - - - 1 1 4.07 * * * - 1 * - * * -

In both tables, the braces below the headings indicate the pairs of cycle sets which are interchanged by the outer automorphism.

Note that [6.35] and [4.04], which have the same configuration, have been put in the first table because they have been checked as giving one another by the transformation; their cycle set allows of self-duality, but the groups are not self-dual.

Not only was [6.07] checked by signature as transforming into itself, but the actual transformed rows were manipulated by transposition and transfigure to coincide exactly with the original rows; this sample check was to allay fears that perhaps a different group had been produced with the same signature.

In his Chapter 12 (p.231) Burnside deals in detail with a process of creating permutation groups isomorphic with a given group. The change ringer tends to think of a group as a permutation group in the first place, but a group may be given more generally as a combination table of its elements, or in geometrical form as set of rotations and/or translations; yet other ways exist in algebra.

Burnside proves that every form of a group as a permutation group arises from the consideration of subgroups of the given group. To explain this in change ringing terms it is necessary to start with a permutation group; as an example take [4.02], the alternating group on 4 bells isomorphic with the rotations of the tetrahedron. The following are its distinct subgroups, each of which will give rise to a distinct representation as a transitive permutation group:

(i) Order 1 Rounds only (ii) Order 2 1234, 2143 [4.07] (iii) Order 3 1234, 2314, 3124 [3.02] (iv) Order 4 1234, 2143, 3412, 4321 [4.04]

Consider first (iii). The subgroup [3.02] is written, together with its cosets, and the four sets are labelled (arbitrarily) 1. to 4.:

1234 2143 3412 4321 2314 1423 4132 3241 3124 4213 1342 2431 1. 2. 3. 4.

To be precise, the columns are transpositions and we create cosets by transfigures (we could equally well do it vice versa). We now transfigure all the cosets by each of the 12 rows of the group in turn. This has the effect of swapping the cosets. The only perm which leaves all the cosets unchanged is rounds. Thus if we apply the transfigure 4132:

4132 1423 3241 2314 1342 4213 2431 3124 3412 2143 4321 1234 3. 2. 4. 1.

This creates a new perm, 3.2.4.1., which itself has 4 figures only because we happened to choose a subgroup of order 3, and 12/3 = 4. In this way, it will be found that the same 4-figure permutation group is re-created, the alternating group on 4 bells.

Next consider (ii). The same process will create six cosets, each containing two perms, and the result will be twelve perms on 6 figures, in fact group [6.14].

Now consider (i). There will be twelve cosets - in fact, the twelve perms of the group - and the result will be a group of twelve perms on 12 bells, which is rather beyond present capabilities in change ringing to utilise!

In the case of (iv) there will be three cosets, numbered 1. to 3. and the result will be the cyclic group 123, 231, 312 four times over. Taken in conjunction with (iii), these create the group [7.33].

It was to be expected that the particular permutation group [4.02] would be re-created, as it was not essential to start the process with a permutation group - other abstract representations of the same mathematical group could have initiated the process - and [4.02] results as one of the alternatives.

Burnside proves that these are the only possible forms of the particular group (the tetrahedral group) as a permutation group. It will be seen from the example of (iv) + (iii) to give [7.33] above that the different forms may be combined into intransitive groups of higher degree, ad infinitum. It is not difficult to find features of the regular tetrahedron corresponding with the degrees of the transitive permutation groups generated above:

12: Corners of faces 6: Edges 4: Vertices, or faces 3: Pairs of opposite edges

I first rationalised this topic when composing two-part peals of Cambridge Surprise Major in 1988. The *universal group* with which I was dealing was the group of all permutations of working bells 23456 (for course-heads), and the group being used was [4.07], i.e. the 2-part form 23456 and 32546. Singles were being considered as well as Bobs, so the universal group was of mixed parity.

The group [4.07] divided the 120 perms into 60 types, and tree searches were performed systematically, starting from different types and excluding types from subsequent searches which had already been "starters". It was noticed that the same peals were being produced starting from different types, and that on this basis one could subdivide the 60 types into 15 sets of 4 types each; moreover, a set of 4 such types expanded into their 8 permutations gave a group of order 8, the dihedral group [4.03].

This phenomenon is explained by the properties of normal subgroups, which are well-documented in mathematical literature. [4.07] is a normal subgroup of [4.03]. Without involving abstract mathematics, we can understand the idea of normal subgroups by examining the distribution of types (*types* being labels for the mathematician's *cosets* which are relevant to normality).

Consider the two groups in question:

(a) Outer group [4.03] order 8, no. of types = 120/8 = 15 (b) Subgroup [4.07] order 2, no. of types = 120/2 = 60

There are 120 perms, which are subdivided by these groups into (a) 15 sets of 8 perms each and (b) 60 sets of 2 perms each. The normality condition to apply is: does each set of 2 perms of the subgroup belong entirely to one particular set of 8 perms of the outer group?

If so, then each type of the outer group (each consisting of 8 perms) must be equivalent to four distinct types of the subgroup (each consisting of 2 of the same 8 perms). It follows that one can draw up a 1:4 correspondence table of types, so that each type of [4.03] is equivalent to a set of four types of [4.07].

Now, any efforts over composition must involve calls such as plain, Bob or Single, which are transpositions for hopping from one type to another. The basic application of the Fundamental Perm Relation, explained early in this paper, ensures that by using group [4.07] any transposition moving from the (two) perms of an arbitrary type a will give the two perms of another type, say type e (exceptionally, types a and e might be identical); we simplify this and say that "type a gives type e". Consider the following extracts:

type x type y group [4.03] +----------------+ +----------------+ | type a type b | | type e type f | group [4.07] | type c type d | | type g type h | +----------------+ +----------------+

Type x of group [4.03] is made up of the four types a, b, c, d of group [4.07]; and similarly type y, of types e, f, g, h. The perms of type a also belong to type x, and those of type e to type y. Thus the chosen transposition moves from a perm in type x to a perm in type y. But [4.03] is itself a group, so that every perm in type x by the chosen transposition must give a perm in type y. Thus types b, c, d of group [4.07] must give types f, g, h (in some unspecified order) by the chosen transposition.

(Groups [4.04], [4.05], [4.06] are all of order 4, and are normal subgroups of [4.03]; while [4.07] is a normal subgroup of them all. The three act as intermediaries in the above, and each splits the four types a, b, c, d into two pairs in a different way)

The consequence of this is, that if a round block consists of a chain of types of group [4.07] containing type a, this type a may be changed to one of the other types within type x (b, c or d) and all the other types in the chain will alter, but each within its own set of 4. Any 2-part round block based on [4.07] may be presented in four different forms as regards the sequence of types in it.

It does not follow that using type a excludes the use of types b, c or d. If they run true with one another, then several or all may be present in a block. But if an exhaustive tree search is made starting from type a, and certain results (or perhaps no results at all) are produced, then it is a waste of time trying b, c or d for starting type because precisely the same results will be produced. By calculating a reduced list of starting types, one from each set of the outer group types, a considerable amount of search time may be saved.

This logic applies in general. Assume we are using a certain group [x.xx] for composition, within a "universal group" of perms (which will usually be either the extent or the +ve half-extent on a certain number of working bells). If there is a group [y.yy] within this universal group, of which [x.xx] is a normal subgroup, then we can expect sets of types of [x.xx] to be equivalent. In the main group listing above, under each group is given a list of "Normal Supergroups". For the example quoted above, we refer to the Normal Supergroups under [4.07], and looking for the mixed group of largest order on up to 5 bells; we find [4.03](8m); m denotes mixed parity, the order is 8, the number of working bells 4. If however we were dealing with just the 60 +ve perms of five bells, the required normal supergroup would be [4.04](4p); whereas if we were in the mixed-parity extent on 6 working bells the relevant normal supergroup would be [6.20](16m). In some situations there are no relevant normal supergroups, consequently all types will give distinct results as "starters"; an example of this is using group [7.05] in the positive half-extent of seven bells, for [7.05] has no positive supergroup.

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